The risk index for an SIR epidemic model and spatial spreading of the infectious disease

Min Zhu; Xiaofei Guo; Zhigui Lin

doi:10.3934/mbe.2017081

Article Contents

2017, Volume 14, Issue 5&6: 1565-1583. Doi: 10.3934/mbe.2017081

This issue Previous Article Threshold dynamics of a time periodic and two–group epidemic model with distributed delay Next Article Modeling and analyzing the transmission dynamics of visceral leishmaniasis

The risk index for an SIR epidemic model and spatial spreading of the infectious disease

1.
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2.
Department of Mathematics, Anhui Normal University, Wuhu 241000, China

* Corresponding author: zglin68@hotmail.com (Z. G. Lin)
* Corresponding author: zglin68@hotmail.com (Z. G. Lin)

Received: May 04, 2016

Accepted: September 18, 2016

Published: November 2017

The first author is supported by Graduate Research and Innovation Projects of Jiangsu Province KYZZ16−0489, and the third author is supported by NSFC of China 11371311 and 11626019.

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Abstract

In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number $R_0^{DA}$ for an associated model with Dirichlet boundary condition, we introduce the risk index $R^F_0(t)$ for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if $R^F_0(t_0)≥q 1$ for some $t_0$ and the disease is vanishing if $R^F_0(∞)<1$, while if $R^F_0(0)<1$, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.

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Mathematics Subject Classification: Primary: 35K51, 35R35; Secondary: 35B40, 92D25.

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Figure 1. $\mu=20$. The left graph shows that the solution $I$ decays to zero quickly. The right graph is the corresponding contour graph, which shows the free boundaries expand slowly and will be limited in a long run

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Figure 2. $\mu=40$. The solution $I$ in the left graph keeps positive and stabilizes to an equilibrium. The right contour graph shows that the free boundaries expand fast

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